{
  "id": "1707.01410",
  "version": "v1",
  "published": "2017-07-05T14:17:18.000Z",
  "updated": "2017-07-05T14:17:18.000Z",
  "title": "Partition algebras $\\mathsf{P}_k(n)$ with $2k>n$ and the fundamental theorems of invariant theory for the symmetric group $\\mathsf{S}_n$",
  "authors": [
    "Georgia Benkart",
    "Tom Halverson"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Assume $\\mathsf{M}_n$ is the $n$-dimensional permutation module for the symmetric group $\\mathsf{S}_n$, and let $\\mathsf{M}_n^{\\otimes k}$ be its $k$-fold tensor power. The partition algebra $\\mathsf{P}_k(n)$ maps surjectively onto the centralizer algebra $\\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ for all $k, n \\in \\mathbb{Z}_{\\ge 1}$ and isomorphically when $n \\ge 2k$. We describe the image of the surjection $\\Phi_{k,n}:\\mathsf{P}_k(n) \\to \\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ explicitly in terms of the orbit basis of $\\mathsf{P}_k(n)$ and show that when $2k > n$ the kernel of $\\Phi_{k,n}$ is generated by a single essential idempotent $\\mathsf{e}_{k,n}$, which is an orbit basis element. We obtain a presentation for $\\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k})$ by imposing one additional relation, $\\mathsf{e}_{k,n} = 0$, to the standard presentation of the partition algebra $\\mathsf{P}_k(n)$ when $2k > n$. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group $\\mathsf{S}_n$. We show under the natural embedding of the partition algebra $\\mathsf{P}_n(n)$ into $\\mathsf{P}_k(n)$ for $k \\ge n$ that the essential idempotent $\\mathsf{e}_{n,n}$ generates the kernel of $\\Phi_{k,n}$. Therefore, the relation $\\mathsf{e}_{n,n} = 0$ can replace $\\mathsf{e}_{k,n} = 0$ when $k \\ge n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-05T14:17:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20C30"
    ],
    "keywords": [
      "partition algebra",
      "symmetric group",
      "invariant theory",
      "fundamental theorems",
      "orbit basis element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}